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Sensor fault estimation filter design for discrete-time linear time-varying systems
Zhenhua Wang, Mickael Rodrigues, Didier Theilliol, Yi Shen
To cite this version:
Zhenhua Wang, Mickael Rodrigues, Didier Theilliol, Yi Shen. Sensor fault estimation filter design for discrete-time linear time-varying systems. Acta Automatica Sinica, 2014, 40 (10), pp.2364-2369.
�10.1016/S1874-1029(14)60365-7�. �hal-00989944�
Sensor Fault Estimation Filter Design for Discrete-Time Linear
Time-Varying Systems
WANG Zhen-Hua1 RODRIGUES Mickael2 THEILLIOL Didier3 SHEN Yi1
Abstract This paper proposes a sensor fault diagnosis method for a class of discrete-time linear time-varying (LTV) systems.
In this paper, the considered system is firstly formulated as a de- scriptor system representation by considering the sensor faults as auxiliary state variables. Based on the descriptor system model, a fault estimation filter which can simultaneously estimate the state and the sensor fault magnitudes is designed via a minimum- variance principle. Then, a fault diagnosis scheme is presented by using a bank of the proposed fault estimation filters. The novelty of this paper lies in developing a sensor fault diagnosis method for discrete LTV systems without any assumption on the dynamic of fault. Another advantage of the proposed method is its ability to detect, isolate and estimate sensor faults in the presence of process noise and measurement noise. Simulation results are given to illustrate the effectiveness of the proposed method.
Key words Fault estimation, linear time-varying (LTV) sys- tems, sensor faults, descriptor system, minimum-variance filter Citation Wang Zhen-Hua, Rodrigues Mickael, Theilliol Didi- er, Shen Yi. Sensor Fault Estimation Filter Design for Discrete- Time Linear Time-Varying Systems. Acta Automatica Sinica, 2014,40(1): 1−3
DOI 10.3724/SP.J.1004.2012.xxxxx
With the growing complexity of modern engineering sys- tems and ever increasing demand for safety and reliability, fault diagnosis techniques have received considerable atten- tion during the past decades. Fruitful results can be found in several excellent monographs[1−3], survey papers[4,5]and the references therein. In the literature, model-based fault detection and isolation (FDI) approaches have been most widely considered. The fundamental idea behind FDI is to generate an alarm when the fault occurs, and then to de- termine the location of fault. However, the fault magnitude cannot be provided by the FDI methods.
Parallel to FDI, the fault estimation methods for dy- namic systems have also been investigated by a number of scholars. Reference [6] proposed an actuator fault es- timation method based on adaptive observer. Adaptive observer-based fault estimation methods for nonlinear sys- tems were studied in [7, 8]. In [9], online learning method- ology was used to deal with fault estimation problem for nonlinear dynamic systems. Most recently, proportional- integral observer has been used to achieve fault estima- tion in descriptor systems[10,11]. However, these aforemen- tioned methods are only studied on fault estimation for continuous-time systems. On the contrary, little attention has been paid to fault estimation in discrete-time systems.
Reference [12] considered fault estimation of actuator faults
Manuscript received Month Date, Year; accepted Month Date, Year Supported by National Natural Science Foundation of China (61273162)
Recommended by Associate Editor BIAN Wei
1. School of Astronautics, Harbin Institute of Technology, Harbin 150001, P. R. China 2. Automatic and Process Control Labo- ratory, University of Lyon, Lyon, F-69003, France; Universit´e Ly- on 1,CNRS UMR 5007, Villeurbanne, F-69622, France 3. Centre de Recherche en Automatique de Nancy, Universit´e de Lorraine, C- NRS UMR 7039, F-54506 Vandoeuvre-les-Nancy, France
for linear multi-input-multi-output systems. In [13], a fault estimation method based on proportional integral observer was proposed for a class of discrete-time nonlinear system- s. However, these methods mainly focus on time-invariant systems with actuator faults. In [14], the authors proposed a fault diagnosis method for discrete-time descriptor linear parameter-varying systems. However, the method in [14]
requires the fault to be constant or slow varying, which is a restrictive condition. [15] proposed anH∞ fault estima- tion method for linear time-varying (LTV) systems based on Krein space approach. However, the control input is not considered in [15].
Recently, references [16, 17] have proposed a descriptor system approach to deal with sensor fault estimation prob- lem. The basic idea behind this method is to construct an augmented descriptor system so that the simultaneous state and fault estimation problem is transformed as the state estimation of the augmented descriptor plant. This methodology provides a novel solution for sensor fault esti- mation. However, [16, 17] only studied sensor fault estima- tion for continuous-time systems. Moreover, process noise and measurement noise are not considered in these works.
In contrast to continuous-time systems, there are few re- sults on sensor fault estimation for discrete-time systems in the literature. Most recently, the authors in [18] have proposed a sensor fault detection, isolation and estimation method for discrete-time Linear Parameter-Varying (LPV) systems. The method in [18] assumes the dynamic of fault can be described by a known model, which is difficult to be determined previously. This assumption restricts the appli- cable scope of the method in [18]. Moreover, if the dynamic model of fault is not properly chosen, it will leads to unde- sirable fault diagnosis results. In [19], a sensor fault estima- tion method for discrete-time systems is proposed by using descriptor Kalman filter. However, [19] only concerns the Linear Time-Invariant (LTI) systems. To the best of our knowledge, sensor fault estimation for discrete-time LTV systems has not been fully investigated, which motivates the presented work.
This paper proposes a sensor fault diagnosis approach for LTV systems. Firstly, a fault estimation filter which can can simultaneously estimate the state and the sensor fault magnitudes is designed by using the descriptor system technique. Then, a fault diagnosis scheme is presented by using a bank of the proposed fault estimation filters. The main contribution lies in two aspects. First, A new fault diagnosis method which is able to detect, isolate and es- timate sensor faults in discrete LTV systems. Compared with the existing result in [19], the proposed method is ap- pliable to LTV systems, which is more challenge to deal with than LTI systems. In comparison with the method in [18], the proposed approach does not make any assump- tion on the dynamic of fault. As a result, the latter has a broader applicable scope and can be used to deal with time-varying faults. Moreover, both process noise and mea- surement noise are considered in this paper, which makes the presented approach practical for real systems.
1 Problem formulation
Consider the following discrete-time LTV system { xk+1=Akxk+Bkuk+Dkwk
yk=Ckxk+Fkfk+vk (1) where xk ∈ Rn, uk ∈ Rp, yk ∈ Rm, wk ∈ Rl and vk ∈ Rm are the state, control input, output, process noise, and measurement noise vectors, respectively. Ak,
2 ACTA AUTOMATICA SINICA Vol. XX Bk,Ck andDk are known matrices of appropriate dimen-
sions. It is assumed thatwkandvkare uncorrelated white noises with covariance matrices Q = E[wkwTk] ≥ 0 and R =E[vkvkT] > 0, respectively. The initial statex0 is of mean ¯x0 and covarianceP0 and is independent ofwk and vk. Fk∈Rm×q represents the sensor fault distribution ma- trix, and the unknown signalfk∈Rqdenotes the effect of the sensor faults. Without loss of generality, it is assumed that matrixFksatisfies
rank (Fk) =q, q < m (2) which implies the the number of faults is less than that of measurements. This assumption is reasonable since the probability for faults to occur at all sensors is very small in practice. It should be noted that the fault distribution matrix Fk is unknown since different fault modes may oc- cur. Therefore, it is reasonable to assume thatFkbelongs to a given set, i.e.
Fk∈ Fk,{
Fk1, Fk2,· · ·, FkM}
(3) Herein,M is the number of possible fault modes.
The main purpose of this paper is to determine the cur- rent fault modeFkand to obtain the estimate for the fault magnitude fk. To this end, this paper proposes a filter synthesis approach to achieve fault estimation for a specif- ic fault mode, and then presents a fault diagnosis scheme based on a bank of dedicated filters.
2 Fault estimation filter design
In this section, a fault estimation filter is designed for a specific fault mode. In this section, the fault distribu- tion matrix is denoted byFk. However, this representation is only used for the convenience of statement because the fault estimation filter synthesis is an essential basis for the fault diagnosis scheme which will be presented in the next section.
To estimate sensor fault, an augmented state vector is defined as
¯ xk=
[xk
fk
]
(4) Then, the system (1) with sensor fault can be written as the following descriptor system
{ E¯xk+1= ¯Akx¯k+ ¯Bkuk+ ¯Dkwk
yk= ¯Ckx¯k+vk (5) where
E= [In 0
0 0
] ,A¯k=
[Ak 0
0 0
] ,B¯k=
[Bk
0 ]
D¯k= [Dk
0 ]
,C¯k=[
Ck Fk]
(6)
If a state estimator is designed for descriptor system (5), then the statexkand sensor faultfk in system (1) can be estimated simultaneously. In other words, by constructing descriptor system (5), simultaneous state and fault esti- mation for system (1) is converted into a state estimation problem of descriptor system (5).
Motivated by the observer proposed in [20], the following filter is constructed for descriptor system (5)
ˆ¯
xk+1=TkA¯kxˆ¯k+TkB¯kuk+Lk(
yk−C¯kxˆ¯k)
+Nkyk+1
(7)
where ˆx¯k∈Rn+q denotes the estimation of the descriptor state ¯xk, Tk ∈ R(n+q)×(n+q), Nk ∈ R(n+q)×m and Lk ∈ R(n+q)×m are matrices to be designed.
In filter (7), matricesTkandNk are designed to satisfy the following equation
TkE+NkC¯k+1=In+q (8) To proceed, we introduce the following Lemma which will be used in the sequel.
Lemma 1. For given matrices B and Y, there exists a matrixX that satisfyX B=Y if and only if
rank [B
Y ]
= rank( Y)
(9) Moreover, a general solution toX B=Y is given by
X =YB†+S(
I− BB†)
(10) whereB† denotes the pseudo-inverse ofBand S is an ar- bitrary matrix.
Proof. Lemma 1 is a straightforward result of the Theorem of Penrose[21].
Since rank (Fk) =q, it is easy to show that rank
[ E C¯k+1
]
=n+q (11)
According to Lemma 1, there exist a matrix[
Tk Nk] satisfying
[Tk Nk] [ E
C¯k+1
]
=In+q (12)
i.e. there exist two matricesTkandNksuch that equation (8) holds.
By using Lemma 1, matricesTk and Nk can be deter- mined by
Tk= Θ†α1+S(
In+q+m−ΘΘ†)
α1 (13) Nk= Θ†α2+S(
In+q+m−ΘΘ†)
α2 (14) whereS∈R(n+q)×(n+q+m)is an arbitrary matrix providing degrees of freedom, matrices Θ ∈ R(n+q+m)×(n+q), α1 ∈ R(n+q+m)×(n+q)andα2∈R(n+q+m)×mare given by
Θ = [ E
C¯k+1 ]
, α1= [In+q
0 ]
, α2= [0
Im ]
(15) For the convenience of statement, the estimation error is denoted as
ek= ¯xk−xˆ¯k (16) and the error covariance matrixPkis defined as
Pk=E[ ekeTk]
(17) Now, the following Theorem is proposed to design matrix Lk in filter (7) by minimizing the trace of the estimation error covariance matrixPk+1=E[
ek+1eTk+1] .
Theorem 1. The gain matrixLk given by Lk=TkA¯kPkC¯kT[
C¯kPkC¯kT+R]−1
(18) minimizes the trace ofPk+1. Moreover, the estimation error covariance matrixPkcan be updated by
Pk+1 = TkA¯kPk( TkA¯k)T
−LkC¯kPk( TkA¯k)T
+TkD¯kQD¯TkTkT+NkRNkT (19) Proof. Using (5) and (7), the error dynamic equation is obtained as follows
ek+1 =(
TkE+NkC¯k+1)
¯
xk+1−xˆ¯k+1
=(
TkA¯k−LkC¯k)
ek+TkD¯kwk
−Lkvk−Nkvk+1
(20) From equation (20), it is easy to derive that
Pk+1 = E[
ek+1eTk+1]
= (
TkA¯k−LkC¯k) Pk(
TkA¯k−LkC¯k)T
+TkD¯kQD¯TkTkT+LkRLTk +NkRNkT
= TkA¯kPk( TkA¯k)T
+TkD¯kQD¯TkTkT+NkRNkT
−LkC¯kPk( TkA¯k)T
−TkA¯kPkC¯kTLTk +Lk(C¯kPkC¯kT+R)
LTk
(21) SinceRis a positive definite matrix, ¯CkPkC¯Tk+Ris also positive definite. Consequently, there exists a nonsingular matrixGk∈Rm×msatisfying
GkGTk = ¯CkPkC¯kT+R (22) Substituting (22) into (21) yields
Pk+1 =TkA¯kPk( TkA¯k)T
+TkD¯kQD¯kTTkT+NkRNkT
−LkC¯kPk( TkA¯k)T
−TkA¯kPkC¯kTLTk +LkGkGTkLTk
(23) Letting
Hk=TkC¯kPkC¯kT(Gk)−1 (24) and substituting (24) into (23) yields
Pk+1 =TkA¯kPk( TkA¯k)T
+TkD¯kQD¯kTTkT+NkRNkT
−LkGkPkHkT−HkGTkLTk +LkGkGTkLTk
=TkA¯kPk( TkA¯k)T
+TkD¯kQD¯kTTkT+NkRNkT
−LkGkPkHkT−HkGTkLTk +LkGkGTkLTk +HkHkT−HkHkT
=TkA¯kPk( TkA¯k)T
+TkD¯kQD¯kTTkT+NkRNkT (LkGk−Hk) (LkGk−Hk)T−HkHkT
(25) From (25), it is obvious that the trace ofPk+1 is mini- mized by letting
LkGk−Hk= 0 (26) Post-multiplying (26) byGTk, it comes
Lk(
C¯kPkC¯kT+R)
−TkA¯kPkC¯kT= 0 (27) Solving equation (27) gives (18).
On the other hand, substituting (26) into (25) gives Pk+1 = TkA¯kPk(
TkA¯k)T
−HkHkT
+TkD¯kQD¯TkTkT+NkRNkT (28)
Using (24) and (22), it can be derived that Pk+1=TkA¯kPk(
TkA¯k)T
+TkD¯kQD¯kTTkT+NkRNkT
−TkA¯kPkC¯kT(
GkGTk)−1C¯kPk( TkA¯k)T
=TkA¯kPk( TkA¯k)T
+TkD¯kQD¯kTTkT+NkRNkT
−TkA¯kPkC¯kT(C¯kPkC¯kT+R)−1C¯kPk( TkA¯k)T
(29) Substituting (18) into (29), we obtain (19).
Remark 1. Although the descriptor system approach has been used to deal with sensor fault estimation prob- lem, most of the existing results focus on continuous-time systems[16,17]. Compared to the existing works, the main contribution of this paper consists in two folds. First, a new sensor fault estimation method for discrete-time LTV systems is proposed. Second, a minimum-variance filter is designed to optimize the fault estimation performance in the presence of process noise and measurement noise.
3 Fault diagnosis scheme
In Section 2, a filter is designed to estimate the sensor faults associated with fault distribution matrixFk. How- ever, different fault mode leads to different fault distribu- tion matrixFk. Therefore, it is also desirable to find out which sensors are faulty when faults have occurred. In this section, we present a fault diagnosis strategy which is sim- ilar to the well-known Dedicated Observer Scheme (DOS).
Fig. 1 illustrates the basic structure of the proposed fault diagnosis scheme.
System fk
Sensor Fault Estimation Filter 1
Sensor Fault Estimation Filteri
Sensor Fault Estimation FilterM uk
ˆ1
xk
Fault Estimation
1
rk
yk
(ˆki) i F
ˆM xk
M
rk
Fault Detection and Isolation
ˆi xk i
rk
ˆ fk
<<
<
<<
<
Figure 1 Basic structure of the fault diagnosis scheme The detail principle of the proposed fault diagnosis scheme is stated in the following.
Since there areMpossible fault modes, all possible faulty models are given as follows
{ xk+1=Akxk+Bkuk+Dkwk
yk=Ckxk+Fkifk+vk , i= 1,· · ·, M (30) Then, the faulty models (30) can be formulated as the fol- lowing descriptor representation
{ E¯xk+1= ¯Akx¯k+ ¯Bkuk+ ¯Dkwk
yk= ¯Cikx¯k+vk , i= 1,· · ·, M (31)
4 ACTA AUTOMATICA SINICA Vol. XX where
E= [In 0
0 0
] ,A¯k=
[Ak 0
0 0
] ,B¯k=
[Bk
0 ]
D¯k= [Dk
0 ]
,C¯ki =[
Ck Fki]
(32)
Then, a bank of sensor fault estimation filters are con- structed as follows
{ xˆ¯ik+1=TkiA¯kxˆ¯k+TkiB¯kuk+Lik(
yk−C¯kixˆ¯ik)
+Nkiyk+1 rki =yk−C¯kixˆ¯ik ,
i= 1,· · ·, M (33) where ˆx¯ikdenotes the augmented state estimation of theith filter, and rik denotes the residual vector of theith filter.
MatricesTki,NkiandLikare designed according to the filter design method proposed in Section 2.
Similar as the methodology in [18], the residual rki can be considered as a quality indicator of theith sensor fault estimation filter. In other words,rki will be close to zero if the ith faulty model is accurate. Otherwise,rki will devi- ate from zero. Given a proper thresholdϵ, we present the following fault detection scheme.
Fault detection scheme: If all ||rik|| ≤ ϵ, i = 1,· · ·, M, then there is no fault. Otherwise, if any||rik||>
ϵ, i= 1,· · ·, M, a fault is detected.
As mentioned before, if theith faulty model is accurate, then rki will be close to zero while rjk, j ̸= iwill deviate from zero. Therefore, the residual corresponding to the actual faulty model exhibits the minimum norm. Based on this observation, the following fault isolation scheme is proposed.
Fault isolation scheme: The fault mode is estimated by
Fˆk=Fki∗ (34) wherei∗is the fault mode index corresponding to the resid- ual with minimum norm, i.e.
i∗= arg
i
i=1,min···, M
{||rki||}
(35) Once the fault mode is estimated, the fault magnitude fkcan be estimated as follows
fˆk=[
0 Iq]xˆ¯ik∗ (36) It is concluded that the sensor faults can be detected, isolated by the proposed fault diagnosis strategy and then be estimated by equation (36).
4 Simulations
In this section, a numerical example is used to illustrate the effectiveness of the proposed method.
Example 1. Consider the discrete-time system in the form of (1) with the following parameters
Ak=
0.2e−k/100 0.6 0 0 0.5 sin (k)
0 0 0.7
, Bk=
1.3 0.5 0.6
Dk=
1 0 0
0 1 0
0 0 1
, Ck=
[1 0 0
0 1 0
]
(37)
and the fault distribution matrixFkbelongs to the follow- ing set
Fk∈ Fk,{
Fk1, Fk2}
(38)
where
Fk1= [1
0 ]
, Fk2= [0
1 ]
(39) In the simulation, the control input isuk= 2sin(0.05k), the initial state is x0 = [0.4 −0.7 0.2]T, and the covari- ance matrices of the process noise and measurement noise sequences areQ= 0.052I3,R= 0.052I2.
In this situation, matrices E, ¯Ak, ¯Bk, ¯Dk, ¯Ck1, ¯Ck2 are determined by (32). A solution to equation (8) is obtained by simply choosing the matrixS in (13) and (14) as
S=
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 1 0 0
Then, by using equations (13), (14), (18) and (19), the gain matricesTk1,Tk2,Nk1,Nk2,L1k,L2kand the variance matrices Pk1,Pk1 can be recursively determined.
To illustrate the performance of the proposed method, the following sensor fault is considered
Fk=Fk2, fk=
{0 k <50
1.2 k≥50 (40)
In this case, ||rk1|| and ||r2k|| are depicted on Fig. 2. It is shown in Fig. 2 that ||rk1|| exceeds the threshold. As a sequence, a sensor fault is detected. It should also be noticed that ||r2k|| is close to zero despite the occurance of fault. In other words, ||r2k||is insensitive to this fault.
Therefore, it can be concluded that the fault mode isFk2, i.e. the second sensor is faulty.
0 10 20 30 40 50 60 70 80 90 100
0 0.2 0.4 0.6 0.8
||r1 k||
0 10 20 30 40 50 60 70 80 90 100
0 0.1 0.2 0.3 0.4
Samples
||r2 k||
Figure 2 The residuals of sensor fault estimation filters in an abrupt fault scenario
Note that the fault mode isFk2, then the fault estimation provided by the 2ed filter is accurate. The fault estimation result is depicted in Fig. 3, where the actual fault is illus- trated by dashed lines and the estimation is represented by solid lines. From Fig. 3, it can be seen that the abrup- t fault can be accurately estimated by the proposed fault estimation filter.
Remark 2. It is noted that the fault estimate previous to 50th sample is zero. The reason is that the fault estimate should be set as zero until a fault occurance is detected.
0 20 40 60 80 100
−0.5 0 0.5 1 1.5 2
Samples
fk
fˆk
Figure 3 The fault estimation result in an abrupt fault scenario
In order to demonstrate the ability of the proposed method in dealing with time-varying faults, the following fault is simulated
Fk=Fk1, fk=
{ 0 k <30
sin(0.2k−6) k≥30 (41) In this situation, ||r1k||and ||r2k||are depicted on Fig. 4.
It is seen that ||r1k|| is close to zero but||rk1||exceeds the threshold. This implies that the first sensor is faulty. In addition, the fault estimation result is depicted in Fig. 5.
From Fig. 5, it can be seen that the fault estimation s- tarts from the 32th sample, which is the fault detection time. This means that the fault detection time-delay is only 2 samples, even if there is a time-varying fault oc- curred. Moreover, Fig. 5 also illustrates that the proposed fault estimation filter exhibits satisfactory performance in estimating time-varying fault.
0 10 20 30 40 50 60 70 80 90 100
0 0.1 0.2 0.3 0.4
||rk1||
0 10 20 30 40 50 60 70 80 90 100
0 0.2 0.4 0.6 0.8 1
Samples
||rk2||
Figure 4 The residuals of sensor fault estimation filters in a time- varying fault scenario
5 Conclusion
This paper proposes a novel sensor fault diagnosis ap- proach for discrete-time LTV systems using descriptor sys- tem technique. The main advantage of the presented method lies in its ability to detect, isolate and estimate sensor faults in the presence of process noise and measure- ment noise. Simulation results show the effectiveness of the proposed method.
0 20 40 60 80 100
−1
−0.5 0 0.5 1 1.5 2
Samples
fk
fˆk
30 32 34
0 0.2 0.4 0.6 0.8
Figure 5 The fault estimation result in a time-varying fault sce- nario
In addition, it should be mentioned that the descriptor system approach merit further research. One of the possible future directions is to extend the results developed in this paper to networked control systems[22,23] or systems with stochastic hybrid dynamics[24−26].
Acknowledgement
The authors would like to acknowledge the anonymous reviewers for their helpful comments and suggestions which improve the presentation of this paper.
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WANG Zhen-Hua Assistant Professor with the Department of Control Science and Engineering, Harbin Institute of Tech- nology. He received his Ph. D. degree from the Department of Control Science and Engineering, Harbin Institute of Technolo- gy in 2013. His research interests include fault diagnosis, fault- tolerant control and observer design.
E-mail: zhwang1987@gmail.com
RODRIGUES Mickael Associate Professor in Claude Bernard University of Lyon 1 with the Automatic and Process Control Laboratory (LAGEP), France. He received his Ph. D.
degree in Automatic Control from the Department of Auto- matic Control of the Henri Poincar´e University of Nancy 1, in 2005 from the Centre de Recherche en Automatique de Nancy (CRAN). His current research interests are focused on model- based fault diagnosis, fault tolerant control, multi-models, LPV systems, singular/descriptor systems, observers, stability and L- MI.
E-mail: mickael.rodrigues@univ-lyon1.fr
THEILLIOL Didier Full Professor in Research Centre for Automatic Control of Nancy (CRAN) at University of Lorraine.
He received his Ph. D. degree in Control Engineering from U- niversity of Lorraine (France) in 1993. His current research in- terests include sensor and actuator model-based fault diagnosis
(FDI) method synthesis and active fault-tolerant control (FTC) system design for LTI, LPV, Multi-linear systems and also reli- ability analysis of components.
E-mail: didier.theilliol@univ-lorraine.fr
SHEN Yi Professor with the Department of Control Science and Engineering, Harbin Institute of Technology. He received his Ph. D. degree from the Department of Control Science and Engineering, Harbin Institute of Technology in 1995. His current research interests include fault diagnosis, flight vehicle control and ultrasound signal processing. Corresponding author of this paper.
E-mail: shen@hit.edu.cn
